Let $X,\{\mathcal F_t\}$ be a weak solution to the SDE $$dX_t=b(X_t)\,dt+\sigma(X_t)\,dWt$$ where $b_i:\Bbb R^d\rightarrow \Bbb R$, $W$ is a $r$-dimensional Brownian motion and $\sigma_{ij}:\Bbb R^d\rightarrow \Bbb R$ for $1\leq i\leq d$, $1\leq j\leq r$.
Let $a_{ik}(x)=\sum_{j=1}^r\sigma_{ij}(x)\sigma_{kj}(x)$ for $1\leq i,k\leq d$, and for $f:\Bbb R^d\rightarrow\Bbb R$ twice continuously differentiable, define $$\mathscr Af(x)=\sum_{i=1}^db_i(x)\frac{\partial f(x)}{\partial x_i}+\frac{1}{2}\sum_{i=1}^d\sum_{k=1}^da_{ik}(x)\frac{\partial^2 f(x)}{\partial x_i\partial x_k}$$
For a progressively measurable and locally integrable process $\{k_t,\mathcal F_t\}$, and $f:\Bbb R^d\rightarrow\Bbb R$ twice continuously differentiable, define $$\Lambda_t:=f(X_t)e^{-\int_0^tk_u\,du}-f(X_0)-\int_0^t(\mathscr Af(X_s)-k_sf(X_s))e^{-\int_0^sk_u\,du}\,ds$$ I am trying to show that $\Lambda,\{\mathcal F_t\}$ is a continuous local martingale (this is Problem 4.34 of Brownian Motion and Stochastic Calculus). I am not sure how I should approach this problem. I can apply Ito on $f(X_t)$ to see that $$M_t:=f(X_t)-f(X_0)-\int_0^t\mathscr Af(X_s)\,ds$$ is a continuous local martingale. However I don't see how to treat the terms such as $f(X_t)e^{-\int_0^tk_u\,du}$ appearing in $\Lambda_t$. I can't use the product rule for this either, because that will require the covariation $[f(X_t),e^{-\int_0^tk_u\,du}]$ which doesn't seem possible to compute. Help would be much appreciated.